Volume of metric balls in Liouville quantum gravity

Abstract

We study the volume of metric balls in Liouville quantum gravity (LQG). For γ ∈ (0, 2), it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for p ∈ (−∞, 4/γ 2 ). Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the γ = 8/3 case. We use this moment bound to show that on a compact set the volume of metric balls of size r is given by r dγ +or (1) , where dγ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Our result implies that the metric measure space structure of γ-LQG determines its conformal structure.